### A CHARACTERIZATION OF TOTALLY REAL CARLEMAN SETS AND AN APPLICATION

### TO PRODUCTS OF STRATIFIED TOTALLY REAL SETS

BENEDIKT S. MAGNUSSON and ERLEND FORNÆSS WOLD

**Abstract**

We give a characterization of stratified totally real sets that admit Carleman approximation by entire functions. As an application we show that the product of two stratified totally real Carleman sets is a Carleman set.

**1. Introduction**

In one complex variable, the so called Carleman sets are well understood: A
closed subset*X*ofCis a Carleman set if and only if (i) it has no interior, (ii)*X*
is polynomially convex, and (iii)C\*X*is locally connected at infinity (Keldych
and Lavrentieff [1]). In the complex plane property (iii) is equivalent to what we
call having bounded E-hulls, but inC* ^{n}*there is no topological characterization.

InC* ^{n}* it is clear that (i)–(iii) is not sufficient for any type of approximation,
as is shown by considering an affine complex line. In particular (i) must be
substituted by some other “lack of complex structure”. In this article we prove
the analogue of the one dimensional result for totally real sets.

Theorem1.1.*LetM* ⊂ ^{C}^{n}*be a closed stratified totally real set. ThenM*
*is a Carleman set if and only ifM* *is polynomially convex and has bounded*
*E-hulls.*

(For the definition of Carleman approximation and bounded E-hulls, see Section 2, and for the definition of a stratified totally real set, see Section 3.)

It is already known [2] that in the totally real setting, the property of bounded E-hulls implies Carleman approximation. The remaining result is therefore the following, which does not rely on the notion of being totally real:

Theorem1.2.*IfM* ⊂^{C}^{n}*is a closed set which admits Carleman approx-*
*imation by entire functions, thenMhas bounded E-hulls.*

Received 25 October 2013.

Theorem 1.2 generalizes the main theorem of [2] where the implication was
shown under the assumption that*M* is totally real and admits*C*^{1}-Carleman
approximation. As an application of this theorem we prove the following partial
answer to a question raised by E. L. Stout (private communication):

Theorem1.3.*LetM** _{j}* ⊂

^{C}

^{n}

^{j}*be stratified totally real sets forj*=1,2

*which*

*admit Carleman approximation by entire functions. ThenM*

_{1}×

*M*

_{2}⊂

^{C}

^{n}^{1}×

^{C}

^{n}^{2}

*admits Carleman approximation by entire functions.*

(For the definition of a stratified totally real set see Section 3.)

The precise question is more general:*ifM*_{j}*are Carleman sets in*C^{n}^{j}*for*
*j* =1,2,*isM*_{1}×*M*_{2}*Carleman?*

A natural generalization of one of the main results of [2] which we will use in the proof of Theorem 1.3 is the following:

Theorem1.4.*LetM* ⊂^{C}^{n}*be a stratified totally real set which has bounded*
*E-hulls, and letK* ⊂ ^{C}^{n}*be a compact set such thatK*∪*M* *is polynomially*
*convex. Then anyf* ∈ *C(K*∪*M)*∩*O(K)is approximable in the Whitney*
*C*^{0}*-topology by entire functions.*

As a corollary to Theorem 1.1 we also obtain the following:

Corollary1.5.*Let* *M* ⊂ ^{C}^{n}*be a totally real manifold of classC*^{k}*and*
*assume thatM* *admits Carleman approximation by entire functions. ThenM*
*admitsC*^{k}*-Carleman approximation by entire functions.*

Proof. By [2] this follows from the fact that*M* has bounded E-hulls.

For more on the topic of Carleman approximation, see*e.g.*the monograph
[5].

**2. Proof of Theorem 1.2**

Deﬁnition2.1. Let*M* ⊂ ^{C}* ^{n}*be a closed set. We say that

*M*is a

*Carleman*

*set, or thatMadmits Carleman approximation by entire functions, ifO(*C

^{n}*)*is dense in

*C(M)*in the Whitney

*C*

^{0}-topology.

Deﬁnition2.2. Let*X*⊂^{C}* ^{n}*be a closed subset. Given a compact normal
exhaustion

*X*

*of*

_{j}*X*we define the

*polynomial hull*of

*X, denoted by*

*X, by*

*X*:= ∪*j**X** _{j}*(this is independent of the exhaustion). We also set

*h(X)*:=

*X*\

*X.*

If*h(X)*is empty we say that*X*is polynomially convex. Note that*X*is closed
and polynomially convex.

Deﬁnition2.3. We say that a closed set*M* ⊂ ^{C}* ^{n}*has

*bounded E-hulls*if for any compact set

*K*⊂

^{C}

*the set*

^{n}*h(K*∪

*M)*is bounded.

We give two lemmas preparing for the proof of Theorem 1.2. The first one is a simple well known result à la Mittag-Leffler and Weierstrass, which we state for the lack of a suitable reference.

Lemma2.4.*Let* *E* = {*x** _{j}*}

*j*∈

^{N}

*be a discrete sequence in*C

^{n}*. Then for any*

*sequence*{

*a*

*}*

_{j}*j*∈

^{N}⊂

^{C}

*there exists an entire functionf*∈

*O(*C

^{n}*)withf (x*

_{j}*)*=

*a*

_{j}*for allj*∈

^{N}

*, and there exist holomorphic functionsf*

_{1}

*, . . . , f*

_{n}*such that*

*E*=

*Z(f*

_{1}

*, . . . , f*

_{n}*).*

Proof. By Theorem 3.7 in [3] there exists an injective holomorphic map
*F* = *(f*˜1*, . . . ,f*˜_{n}*):*C* ^{n}* →

^{C}

*such that*

^{n}*F (x*

_{j}*)*=

*j*·

**e**1. For the first claim we let

*g*∈

*O(*C

*)*be an entire function with

*g(j)*=

*a*

*for all*

_{j}*j*∈

^{N}and set

*f*=

*g*◦ ˜

*f*

_{1}. For the second claim let

*g*∈

*O(*C

*)*be an entire function whose zero set is precisely{

*j*}

*j*∈

^{N}. Now set

*f*

_{1}=

*g*◦ ˜

*f*

_{1}and

*f*

*= ˜*

_{k}*f*

*for*

_{k}*k*=2, . . . , n.

Lemma 2.5.*Let* *M* ⊂ ^{C}^{n}*be a Carleman set and letE* = {*x** _{j}*}

*j*∈N

*be a*

*discrete set of points withE*⊂

^{C}

*\*

^{n}*M. ThenM*∪

*Eis a Carleman set, with*

*interpolation onE.*

Proof. Assume that*q:M*∪*E*→^{C}and*ε:M*∪*E*→^{R}+are continuous
functions. By Lemma 2.4 there exist functions*f*_{1}*, . . . , f** _{n}* ∈

*O(*C

^{n}*)*such that

*f*

_{j}*(x*

_{k}*)*=0 for all

*x*

*∈*

_{k}*E*and

*j*=1, . . . , n, and such that

*Z(f*1

*, . . . , f*

_{n}*)*∩

*M*=

∅. So there exist continuous functions*g** _{j}* ∈

*C(M)*such that

*g*

_{1}·

*f*

_{1}+ · · · +

*g*

*·*

_{n}*f*

*=1*

_{n}on*M*. Since*M*admits Carleman approximation we may approximate the*g** _{j}*’s
by entire functions

*g*˜

*∈*

_{j}*O(*C

^{n}*)*such that the function

*ϕ*= ˜*g*1·*f*1+ · · · + ˜*g**n*·*f**n*

satisfies*ϕ(x)* =0 for all*x*∈*M. Obviouslyϕ(z)*=0 for all*z*∈*E.*

By the Mittag-Leffler Theorem there exists an entire function*h* ∈ *O(*C^{n}*)*
such that*h(z)*= *q(z)*for all*z* ∈*E. Letψ* ∈*C(M)*be the function*ψ(x)*:=

*h(x)*−*q(x)*

*ϕ(x)* . Since*M* admits Carleman approximation we may approximate*ψ*
by en entire function*σ* ∈ *O(*C^{n}*), and if the approximation is good enough,*
the function*h(z)*−*ϕ(z)*·*σ (z)*is*ε-close toq*on*M*∪*E.*

Proof of Theorem 1.2. Aiming for a contradiction we assume that*M*
does not have bounded exhaustion hulls,*i.e., there exists a compact setK*such
that*h(K*∪*M)*is not bounded. This implies that there is a discrete sequence of
points*E*= {*x** _{j}*} ⊂

*h(K*∪

*M). By Lemma 2.5 there exists an entire function*

*q*∈

*O(*C

^{n}*)*such that|

*q(z)*|

*<*

^{1}

_{2}, for

*z*∈

*M*and

*q(x*

_{j}*)*=

*j*for

*x*

*∈*

_{j}*E. Define*

*C*=

*q*

*K*. For

*j > C*we then have that|

*q(x*

_{j}*)*|

*>*sup

_{z}_{∈}

_{K}_{∪}

*{|*

_{M}*q(z)*|}which contradicts the assumption that

*E*⊂

*h(K*∪

*M).*

**3. Proof of Theorem 1.4**

Deﬁnition3.1. Let *M* ⊂ ^{C}* ^{n}* be a closed set. We say that

*M*is a

*stratified*

*totally real set*if

*M*is the increasing union

*M*

_{0}⊂

*M*

_{1}⊂ · · · ⊂

*M*

*=*

_{N}*M*of closed sets, such that

*M*

*\*

_{j}*M*

_{j}_{−}

_{1}is a totally real set (a set which is locally contained in a totally real manifold) for

*j*= 1, . . . , N, and with

*M*0totally real.

The proof of Theorem 1.4 is an inductive construction depending on the following lemma [4, Theorem 4.5]

Lemma 3.2. *Let* *K* ⊂ ^{C}^{n}*be a compact set, letM* ⊂ ^{C}^{n}*be a compact*
*stratified totally real set, and assume thatK*∪*Mis polynomially convex. Then*
*any functionf* ∈ *C(K*∪*M)*∩*O(K)is uniformly approximable by entire*
*functions.*

Proof. Set *X** _{j}* :=

*K*∪

*M*

*for*

_{j}*j*= 0, . . . , N. Then

*X*

_{0}is polynomially convex (see the proof of Theorem 4.5 in [4]) and so it follows from [2] that

*f*|

*X*0

is uniformly approximable by entire functions. The result is now immediate from Theorem 4.5 in [4].

Proof of Theorem1.4. Choose a normal exhaustion*K** _{j}* ofC

*such that*

^{n}*K*

*∪*

_{j}*M*is polynomially convex for each

*j*. Assume that we are given

*f*∈

*C(K*∪

*M)*∩

*O(K),ε*∈

*C(K*∪

*M)*with

*ε(x) >*0 for all

*x*∈

*K*∪

*M*. Set

*K*

_{0}=

*K*

_{1}=

*K,f*

_{0}=

*f*

_{1}=

*f*. We will construct an approximation of

*f*by induction on

*j*, and we assume that, for

*k*= 1, . . . , j, we have constructed

*f*

*∈*

_{k}*C(K*

*∪*

_{k}*M)*∩

*O(K*

_{k}*)*such that

*(1*_{k}*)* |*f*_{k}*(x)*−*f (x)*|*< ε(x)/2* for all *x*∈*M,*
and

*(2*_{k}*)* *f** _{k}*−

*f*

_{k}_{−}1

*K*

*k−1*

*< (1/2)*

^{k}*.*

Choose*χ** _{j}* ∈

*C*0

^{∞}

*(K*

_{j}^{◦}

_{+}

_{2}

*)*with

*χ*

*≡1 near*

_{j}*K*

_{j}_{+}1. For any 0

*< δ*

_{j}*< (1/2)*

^{j}^{+}

^{1}it follows from Lemma 3.2 that there exists an entire function

*g*

*such that*

_{j}|*g*_{j}*(x)*−*f*_{j}*(x)*|*< δ** _{j}* for all

*x*∈

*K*

*∪*

_{j}*(M*∩

*K*

_{j}_{+}

_{2}

*).*

We define*f*_{j}_{+}_{1}:=*χ** _{j}*·

*g*

*+*

_{j}*(1*−

*χ*

_{j}*)(f*

_{j}*)*on

*M*, and

*f*

_{j}_{+}

_{1}:=

*g*

*near*

_{j}*K*

_{k}_{+}

_{1}. It is clear that we get

*(2*_{j}_{+}_{1}*)* *f*_{j}_{+}_{1}−*f*_{j}*K**j* *< δ*_{j}*< (1/2)*^{j}^{+}^{1}*,*
and on*M*\*K** _{j}* we have that

*f*_{j}_{+}_{1}−*f* =*(f** _{j}*−

*f )*+

*χ*

*·*

_{j}*(g*

*−*

_{j}*f*

_{j}*),*

so if*δ** _{j}* is sufficiently small we also get that

*(1*_{j}_{+}_{1}*)* |*f*_{j}_{+}_{1}*(x)*−*f (x)*|*< ε(x)/2* for all *x*∈*M.*

It follows from*(2*_{k}*)*that*f** _{k}* converges to an entire function

*f*˜, and it follows from

*(1*

_{k}*)*that| ˜

*f (x)*−

*f (x)*|

*< ε(x)*for all

*x*∈

*K*∪

*M*.

**4. Proof of Theorem 1.3**

Note first that*M*_{1}×*M*_{2}is a stratified totally real set which is polynomially
convex, so by Theorem 1.4 it suffices to show that*M*1×*M*2 has bounded
E-hulls. Let*K*⊂^{C}^{n}^{1}×^{C}^{n}^{2}be compact. Since both*M*1and*M*2are Carleman
sets, they have bounded exhaustion hulls by Theorem 1.2. Choose compact
sets*K*˜* _{j}* inC

^{n}*for*

^{j}*j*=1,2,with

*K*⊂ ˜

*K*1× ˜

*K*2. Set

*K*

*:= ˜*

_{j}*K*

*∪*

_{j}*h(K*˜

*∪*

_{j}*M*

_{j}*)*which are now compact sets.

We claim that*(K*_{1}×*K*_{2}*)*∪*(M*_{1}×*M*_{2}*)*is polynomially convex, from which
it follows that*h(K* ∪*(M*1×*M*2*))*⊂ *K*1×*K*2. Let*(z*0*, w*0*)*∈*(*C^{n}^{1} ×^{C}^{n}^{2}*)*\
[(K_{1}×*K*_{2}*)*∪*(M*_{1}×*M*_{2}*)]. We consider several cases.*

(i) *z*_{0} ∈*/* *K*_{1}∪*M*_{1}. Here,*K*_{1}∪*M*_{1}is polynomially convex and we simply
use a function in the variable*z*_{0}only.

(ii) *w*_{0}∈*/K*_{2}∪*M*_{2}. Analogous to (i).

(iii) *z*_{0}∈*M*_{1}∩*K*_{1}. Then*w*_{0}∈*/K*_{2}∪*M*_{2}, so we are in case (ii).

(iv) *z*_{0} ∈ *M*_{1} \*K*_{1}. Then *w*_{0} ∈ *K*_{2}\*M*_{2}, unless we are in case (ii). By
Lemma 2.5 there exists*f* ∈*O(*C^{n}^{2}*)*such that*f (w*_{0}*)*=1 and|*f (w)*|*<*

1/2 for all *w* ∈ *M*_{2}. Set *N* = *f**K*2. By Theorem 1.4 there exists
*g* ∈*O(*C^{n}^{1}*)*such that*g**K*1 *<*1/(2N),|*g(z)*|*<*3/2 for all*z*∈*M*_{1}and
*g(z*0*)*=1. Set*h(z, w)*=*f (w)*·*g(z). Thenh(z*0*, w*0*)*=1. For*(z, w)*∈
*K*_{1}×*K*_{2}we have|*h(z, w)*| = |*f (w)*||*g(z)*| ≤ *N* ·1/(2N) = 1/2. If
*(z, w)*∈*M*_{1}×*M*_{2}then|*h(z, w)*| = |*f (w)*||*g(z)*| ≤1/2·3/2=3/4.

(v) *z*_{0}∈*K*_{1}\*M*_{1}. Then*w*_{0}∈*M*_{2}\*K*_{2}unless we are in case (ii), but this is
the same as (iv) with the roles of*z*_{0}and*w*_{0}switched.

REFERENCES

1. Keldych, M., and Lavrentieff, M.,*Sur un problème de M. Carleman, C. R. (Doklady) Acad.*

Sci. URSS (N. S.) 23 (1939), 746–748.

2. Manne, P. E., Wold, E. F., and Øvrelid, N.,*Holomorphic convexity and Carleman approxim-*
*ation by entire functions on Stein manifolds, Math. Ann. 351 (2011), no. 3, 571–585.*

3. Rosay, J.-P., and Rudin, W.,*Holomorphic maps from*C^{n}*to*C* ^{n}*, Trans. Amer. Math. Soc. 310
(1988), no. 1, 47–86.

4. Samuelsson, H., and Wold, E. F.,*Uniform algebras and approximation on manifolds, Invent.*

Math. 188 (2012), no. 3, 505–523.

5. Stout, E. L.,*Polynomial convexity, Progress in Mathematics, vol. 261, Birkhäuser Boston,*
Inc., Boston, MA, 2007.

SCIENCE INSTITUTE UNIVERSITY OF ICELAND DUNHAGA 5

107 REYKJAVIK ICELAND

*E-mail:*bsm@hi.is

DEPARTMENT OF MATHEMATICS UNIVERSITY OF OSLO P.O. BOX 1053 BLINDERN NO-0316 OSLO NORWAY

*E-mail:*erlendfw@math.uio.no